Studies in Systems, Decision and Control 254 Tofigh Allahviranloo Uncertain Information and Linear Systems Studies in Systems, Decision and Control Volume 254 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and withahighquality.Theintentistocoverthetheory,applications,andperspectives on the state of the art and future developments relevant to systems, decision making,control,complexprocessesandrelatedareas, asembeddedinthefieldsof engineering,computerscience,physics,economics,socialandlifesciences,aswell astheparadigmsandmethodologiesbehindthem.Theseriescontainsmonographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. 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More information about this series at http://www.springer.com/series/13304 fi To gh Allahviranloo Uncertain Information and Linear Systems 123 TofighAllahviranloo FacultyofEngineeringandNaturalSciences Bahçeşehir University Istanbul,Turkey ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems,DecisionandControl ISBN978-3-030-31323-4 ISBN978-3-030-31324-1 (eBook) https://doi.org/10.1007/978-3-030-31324-1 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Jahanshahloo Preface In this book, I tried to introduce and apply the uncertain information or data in severaltypes toanalyzethelinearsystems.Theseversionsofinformationarevery applicable in our applied science. The initial subjects of this book point out the important uncertainties to use in real-life problem modeling. Having information about several types of ambiguities, vagueness, and uncer- tainties is important in modeling the problems that involve linguistic variables, parameters, and word computing. Nowadays, most of our real-life problems are relatedtodecisionmakingattherighttime,andtherefore,weshoulduseintelligent decision science. Clearly, every intelligent system needs real data in our environ- ment to have an appropriate and flexible mathematical model. Most of the men- tioned problems can be modeled by mathematical models, and a system of linear equationsistheirfinalstatusthatmustbesolved.Thenewestversionsofuncertain information have been discussed in this book. This book has been prepared for all undergraduate students in mathematics, computerscience,andengineeringinvolvedwithfuzzyanduncertainty.Especially in industrial engineering and applied mathematics in the field of optimization, one of the most important subjects is the linear systems with uncertainty. Istanbul, Turkey Tofigh Allahviranloo September, 2019 vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction to Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Distribution Functions . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Measurable Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.3 Uncertainty Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2.4 Uncertainty Distribution Functions . . . . . . . . . . . . . . . . 16 2.2.5 Uncertain Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.6 Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2.7 Level Wise Membership Function or Interval Form. . . . 31 2.2.8 Arithmetic on Intervals Form of Membership Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.9 Distance Between Uncertain Sets . . . . . . . . . . . . . . . . . 49 2.2.10 Ranking of Uncertain Sets . . . . . . . . . . . . . . . . . . . . . . 56 3 Uncertain Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2 Uncertain Vector and Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 The Solution Set of an Uncertain Linear System . . . . . . . . . . . . 68 3.4 Solution Sets of Uncertain System of Linear Equations in Interval Parametric Format. . . . . . . . . . . . . . . . . . . 70 3.5 The System of Linear Equations with Uncertain RHS. . . . . . . . . 92 3.6 Uncertain Complex System. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.7 An Approach to Find the Algebraic Solution for Systems with Uncertain RHS. . . . . . . . . . . . . . . . . . . . . . . . 119 3.8 An Estimation of the Solution of an Uncertain Systems with Uncertain RHS . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ix x Contents 3.8.1 Interval Gaussian Elimination Method. . . . . . . . . . . . . . 143 3.9 Allocating Method for the Uncertain Systems with Uncertain RHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.10 Allocating Method for the Fully Uncertain Systems . . . . . . . . . . 163 3.10.1 Allocating Method for the Fully Uncertain Systems (Non-symmetric Solutions). . . . . . . . . . . . . . . . . . . . . . 173 3.11 LR Solution for Systems with Uncertain RHS (Best Approximation Method). . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.12 LR Solution for Systems with Uncertain RHS (Distance Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4 Advanced Uncertainty and Linear Equations. . . . . . . . . . . . . . . . . . 211 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.2 The Uncertain Arithmetic on Pseudo-octagonal Uncertain Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.2.1 The Uncertain Arithmetic Operations on Pseudo-octagonal Uncertain Sets . . . . . . . . . . . . . . . 213 4.2.2 Solving Uncertain Equation as A þ X ¼B. . . . . . . . . . 229 4.2.3 Solving Uncertain Equation as A(cid:2)X ¼B . . . . . . . . . . . 230 4.2.4 Solving Uncertain Equation as A(cid:2)X þ B¼C . . . . . . . 232 4.3 Combined Uncertain Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 4.3.1 Ranking of Combined Uncertain Sets . . . . . . . . . . . . . . 245 4.3.2 Distance Between Combined Uncertain Sets . . . . . . . . . 246 4.3.3 Ranking Method Based on Expected Value. . . . . . . . . . 247 4.3.4 Advanced Combined Uncertain Sets (ACUSs). . . . . . . . 250 Bibliography .. .... .... .... ..... .... .... .... .... .... ..... .... 255 Chapter 1 Introduction 1.1 Introduction Let’s start with a sentence from ‘Albert Einstein’: Asforthelawsofmathematicsrefertoreality,theyarenotcertain,andasfarastheyare certain,theydonotrefertoreality. Since the mathematical laws point to reality, this point is not conjectured with certainty,andsinceitspeaksdecisivemathematicalrules,itdoesnotrefertoreality and is far from reality. In fact, uncertainty has a history of human civilization and humanity has long been thinking of controlling and exploiting this type of information. One of the most ancient and obscure concepts has been the phrase “luck”. Evidence of gam- bling is said to have been obtained in Egypt in 3500 BC and found similar to the current dice there. The gambling and dice have acted an important role in devel- oping the theory of probability. In the 15th century, Gerolamo Cardano was one of the most knowledgeable individualsinthefieldofformaloperationsofalgebra.Inhis“GameofChance”,he presented his first analysis of lucky laws. In this century, Galileo Galilei has also solved such problems in numerical form. In 1657, Christiaan Huygens wrote the first book on probability entitled “On the calculation of chance games.” This book was a real birthday of probability. The theory of probability started mathematically by Blaise Pascal and Pierre de Fermatinthe17thcentury,whichsoughttosolvemathematicalproblemsincertain gambling issues. From the seventeenth century, the theory of probability was constantly devel- oped and applied in various disciplines. Nowadays, the possibility in most engi- neering and management fields is an important tool, and even its use in medicine, ethics, law, and so on. In this regard, Pascal says: It’s great that science was ©SpringerNatureSwitzerlandAG2020 1 T.Allahviranloo,UncertainInformationandLinearSystems, StudiesinSystems,DecisionandControl254, https://doi.org/10.1007/978-3-030-31324-1_1